Finite Group Databasebeta

← #10696939

Subgroups — #10696939

Order 768.

604 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic, complemented
3#83159C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#10147—31nomaximal, Sylow
9C22×C3×SD16—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#492—22no—
14C3×#492—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5228—22no—
17#5215—22no—
18#5092—21no—
19#5090—22no—
20#5079—21no—
21#5071—22no—
22#5070—21no—
23#5070—21no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4544—22no—
28#4530—22no—
29#4456—22no—
30#3209—31no—
31C2×#492D311yescharacteristic
32#3138—31no—
33#1997—31no—
34#1545—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8D412yes—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C2×C3×SD16—28no—
42C2×C3×SD16—48no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127D412yes—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#216—24no—
56#926—21no—
57#924—21no—
58#924—21no—
59C2×#208—21no—
60C2×#208D412yes—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4×C8—21no—
67C3×C4×C8D412yes—
68#835—24no—
69#832—24no—
70#832—24no—
71#830—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C22×SD16—22no—
79#533—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127C2×D311yescharacteristic
82#494—61no—
83#492—22no—
84#492—22no—
85#490—62no—
86#487—61no—
87#486—61no—
88#486—61no—
89#462—62no—
90#451—62no—
91#431—31no—
92#423—31no—
93#418—62no—
94C4×M16—31no—
95C2×C4×C8C2×D311yescharacteristic
96#343—62no—
97#331—62no—
98Q8⋊C8—62no—
99C24×C3—41no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×D4—22no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—44no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—42no—
111C2×C3×D4—42no—
112C2×C3×D4—82no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×SD16—416no—
120C2×C3×C8—24no—
121C2×C3×C8—22no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#217—41no—
131#216—84no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—22no—
140#208—22no—
141#208—42no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8#6712yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C2×Q16—64no—
150C2×SD16—28no—
151C2×SD16—48no—
152C2×M16—32no—
153C2×M16—61no—
154C22×C8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161#127—21no—
162#127—21no—
163#127—21no—
164#127#6712yes—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C42C22×D311yescharacteristic
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#103—64no—
178#102—64no—
179#102—64no—
180C22⋊C8—61no—
181#97—61no—
182C4×C8—21no—
183C4×C8—61no—
184C4×C8—61no—
185C4×C8#6712yes—
186C23×C3—42no—
187C23×C3—82no—
188C23×C3—82no—
189C23×C3—82no—
190C23×C3#12111yescharacteristic
191C3×Q8—44no—
192C3×Q8—44no—
193C3×D4—44no—
194C3×D4—44no—
195C3×D4—82no—
196C3×D4—82no—
197C3×D4—82no—
198C3×D4—81no—
199C3×D4—81no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×C3×C4#13512yes—
211C2×C3×C4#13712yes—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—42no—
215Dic6—42no—
216C3×C8—24no—
217C3⋊C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C24—41no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—44no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—82no—
236C22×C4—22no—
237C22×C4D3×D411yescharacteristic
238C22×C4#23711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240Q16—244no—
241SD16—416no—
242M16—62no—
243M16—62no—
244M16—122no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—62no—
248C2×C8—62no—
249C2×C8—121no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C22⋊C4—121no—
261C42—121no—
262C42D3×D411yescharacteristic
263C42D3×D411yescharacteristic
264C42C2×#6711yescharacteristic
265C42C2×#6711yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—82no—
269C22×C3—82no—
270C22×C3—82no—
271C22×C3—164no—
272C22×C3#39211yescharacteristic
273C22×C3#46412yes—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4#45212yes—
278C3×C4#45312yes—
279C3×C4#47112yes—
280C3×C4#49511yescharacteristic
281C3×C4#49711yescharacteristic
282Dic3—81no—
283Dic3—81no—
284C23—42no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23#93911yescharacteristic
289Q8—44no—
290Q8—44no—
291Q8—122no—
292Q8—122no—
293D4—44no—
294D4—44no—
295D4—82no—
296D4—81no—
297D4—81no—
298D4—82no—
299D4—82no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4#91512yes—
307C2×C4#91612yes—
308C2×C4#93711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4#95311yescharacteristic
314C8—24no—
315C8—122no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—162no—
320C2×C3#200211yescharacteristic
321C2×C3#201511yescharacteristic
322C2×C3#201911yescharacteristic
323C22—21no—
324C22—21no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—164no—
329C22#516612yes—
330C22#522311yescharacteristic
331C4—21no—
332C4—21no—
333C4—82no—
334C4—241no—
335C4—241no—
336C4#507712yes—
337C4#516711yescharacteristic
338C4#516811yescharacteristic
339C4#517312yes—
340C4#517712yes—
341C3#1014711yescharacteristic, complemented, Sylow
342C2—22no—
343C2—162no—
344C2#7176911yescharacteristic
345C2#7205811yescharacteristic
346C2#7205911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#492C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenterC2×#492D3
hypocenterC3#10147
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#492

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #83222C2semidirect
4. #71790C2semidirect
6. #69738C2semidirect
7. #67656C2semidirect
341. C3#10147semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#4922
#832222
#831592
#717902
#717592
#697382
#676562
#101473